"With the ball being where it was and with Ichiro out there, I didn't think we had much of a chance," Zobrist said.
But Suzuki's throw was off line and missed the cutoff man, forcing catcher Jesus Montero to field the ball on the first-base side of the plate.
Montero scrambled to get into position to make the tag, but Pena just got in under it.
"Missing the cutoff permitted that to happen," Rays manager Joe Maddon said. "We got very fortuitous that particular moment."
The Rays went 2-for-15 with runners in scoring position, and stranded the potential tying runs in those situations in the 10th, 11th and 12th innings.
B.J. Upton homered and drove in two runs for Tampa Bay.
Pena singled off Tom Wilhelmsen with one out in the 14th, and Zobrist followed with his third hit of the game.
"It's kind of a relief when you win a game like that," said Zobrist, who walked twice and was hit by a pitch.
Jake McGee (3-2) pitched two-thirds of an inning to escape a jam in the top of the 14th and got the win. The run off Wilhelmsen (3-2) ended the reliever's streak of 24 consecutive scoreless innings over 20 appearances.
Carlos Peguero hit his third homer in five career at-bats off James Shields, a two-run shot that gave the Mariners a 3-2 lead in the seventh. He went deep twice against the Rays right-hander at Safeco Field on June 2, 2011, and hadn't homered in a major league game since hitting one against the Miami Marlins later that month.
Upton homered off Mariners starter Hisashi Iwakuma in the sixth inning, then drove in a second run with an infield single to make it 3-all in the bottom of the seventh.
Shields allowed three runs and four hits over 7 2/3 innings, ending a stretch of four consecutive starts in which he yielded at least 10 hits -- the longest such streak in the American League since Dan Haren did it with Oakland in 2007. No Rays pitcher had done it since Tanyon Sturtze in 2002.
Rays pitchers combined to strike out 17. Six relievers held the Mariners to one hit over the final 6 1/3 innings.
Meanwhile, eight Seattle pitchers fanned 15. The Rays had three hits after Upton tied it in the seventh.
"We just weren't able to get anything going offensively in extra innings," Mariners manager Eric Wedge said. "Our pitching staff did a great job."
Suzuki tripled and Casper Wells doubled to give the Mariners a 1-0 lead in the first. Shields hit a batter and walked Kyle Seager to load the bases before escaping further damage by striking out Justin Smoak.
Shields settled down after the shaky beginning to retire 12 straight after the walk to Seager with two outs in the first. The Mariners didn't have another runner until shortstop Sean Rodriguez mishandled Dustin Ackley's ground ball for an error with two outs in the fifth.
Iwakuma held the Rays scoreless through five innings before giving up two runs in the sixth on Upton's ninth homer and Jeff Keppinger's two-out RBI hit that deflected off third baseman Seager's glove to briefly put Tampa Bay ahead 2-1.
Seattle regained the lead with the first hits off Shields since the first inning. Seager singled leading off the seventh, and Peguaro followed with first homer since being promoted from Triple-A Tacoma on July 5.
Shields allowed fewer than four runs in a game for the first time since giving up three runs and eight hits over 7 2/3 innings while also not being involved in the decision of a Tampa Bay loss to Miami on June 16. He walked two and struck out 10 before being replaced by Joel Peralta with two outs and a runner on first base in the eighth.
Fernando Rodney pitched the ninth and 10th for the Rays.
Upton turns 28 on Aug. 21. The center fielder needs one more homer before his birthday to join Rickey Henderson, Barry Bonds, Bobby Bonds, Cesar Cedeno, Lloyd Moseby, Eric Davis and Hanley Ramirez as the only players to amass 100 homers and 200 steals before age 28. ... Seattle RHP Erasmo Ramirez (right elbow) is scheduled for a simulated game Saturday and could be ready for a minor league rehab assignment. ... Mariners INF Franklin Gutierrez (concussion) is with the team and working out. ... Rays OF Sam Fuld, who hasn't played this season while recovering from right wrist surgery, might be close to returning from a rehabilitation assignment at Triple-A Durham.